1*> \brief \b ZPBSTF 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download ZPBSTF + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zpbstf.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zpbstf.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zpbstf.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE ZPBSTF( UPLO, N, KD, AB, LDAB, INFO ) 22* 23* .. Scalar Arguments .. 24* CHARACTER UPLO 25* INTEGER INFO, KD, LDAB, N 26* .. 27* .. Array Arguments .. 28* COMPLEX*16 AB( LDAB, * ) 29* .. 30* 31* 32*> \par Purpose: 33* ============= 34*> 35*> \verbatim 36*> 37*> ZPBSTF computes a split Cholesky factorization of a complex 38*> Hermitian positive definite band matrix A. 39*> 40*> This routine is designed to be used in conjunction with ZHBGST. 41*> 42*> The factorization has the form A = S**H*S where S is a band matrix 43*> of the same bandwidth as A and the following structure: 44*> 45*> S = ( U ) 46*> ( M L ) 47*> 48*> where U is upper triangular of order m = (n+kd)/2, and L is lower 49*> triangular of order n-m. 50*> \endverbatim 51* 52* Arguments: 53* ========== 54* 55*> \param[in] UPLO 56*> \verbatim 57*> UPLO is CHARACTER*1 58*> = 'U': Upper triangle of A is stored; 59*> = 'L': Lower triangle of A is stored. 60*> \endverbatim 61*> 62*> \param[in] N 63*> \verbatim 64*> N is INTEGER 65*> The order of the matrix A. N >= 0. 66*> \endverbatim 67*> 68*> \param[in] KD 69*> \verbatim 70*> KD is INTEGER 71*> The number of superdiagonals of the matrix A if UPLO = 'U', 72*> or the number of subdiagonals if UPLO = 'L'. KD >= 0. 73*> \endverbatim 74*> 75*> \param[in,out] AB 76*> \verbatim 77*> AB is COMPLEX*16 array, dimension (LDAB,N) 78*> On entry, the upper or lower triangle of the Hermitian band 79*> matrix A, stored in the first kd+1 rows of the array. The 80*> j-th column of A is stored in the j-th column of the array AB 81*> as follows: 82*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 83*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 84*> 85*> On exit, if INFO = 0, the factor S from the split Cholesky 86*> factorization A = S**H*S. See Further Details. 87*> \endverbatim 88*> 89*> \param[in] LDAB 90*> \verbatim 91*> LDAB is INTEGER 92*> The leading dimension of the array AB. LDAB >= KD+1. 93*> \endverbatim 94*> 95*> \param[out] INFO 96*> \verbatim 97*> INFO is INTEGER 98*> = 0: successful exit 99*> < 0: if INFO = -i, the i-th argument had an illegal value 100*> > 0: if INFO = i, the factorization could not be completed, 101*> because the updated element a(i,i) was negative; the 102*> matrix A is not positive definite. 103*> \endverbatim 104* 105* Authors: 106* ======== 107* 108*> \author Univ. of Tennessee 109*> \author Univ. of California Berkeley 110*> \author Univ. of Colorado Denver 111*> \author NAG Ltd. 112* 113*> \ingroup complex16OTHERcomputational 114* 115*> \par Further Details: 116* ===================== 117*> 118*> \verbatim 119*> 120*> The band storage scheme is illustrated by the following example, when 121*> N = 7, KD = 2: 122*> 123*> S = ( s11 s12 s13 ) 124*> ( s22 s23 s24 ) 125*> ( s33 s34 ) 126*> ( s44 ) 127*> ( s53 s54 s55 ) 128*> ( s64 s65 s66 ) 129*> ( s75 s76 s77 ) 130*> 131*> If UPLO = 'U', the array AB holds: 132*> 133*> on entry: on exit: 134*> 135*> * * a13 a24 a35 a46 a57 * * s13 s24 s53**H s64**H s75**H 136*> * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54**H s65**H s76**H 137*> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 138*> 139*> If UPLO = 'L', the array AB holds: 140*> 141*> on entry: on exit: 142*> 143*> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 144*> a21 a32 a43 a54 a65 a76 * s12**H s23**H s34**H s54 s65 s76 * 145*> a31 a42 a53 a64 a64 * * s13**H s24**H s53 s64 s75 * * 146*> 147*> Array elements marked * are not used by the routine; s12**H denotes 148*> conjg(s12); the diagonal elements of S are real. 149*> \endverbatim 150*> 151* ===================================================================== 152 SUBROUTINE ZPBSTF( UPLO, N, KD, AB, LDAB, INFO ) 153* 154* -- LAPACK computational routine -- 155* -- LAPACK is a software package provided by Univ. of Tennessee, -- 156* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 157* 158* .. Scalar Arguments .. 159 CHARACTER UPLO 160 INTEGER INFO, KD, LDAB, N 161* .. 162* .. Array Arguments .. 163 COMPLEX*16 AB( LDAB, * ) 164* .. 165* 166* ===================================================================== 167* 168* .. Parameters .. 169 DOUBLE PRECISION ONE, ZERO 170 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) 171* .. 172* .. Local Scalars .. 173 LOGICAL UPPER 174 INTEGER J, KLD, KM, M 175 DOUBLE PRECISION AJJ 176* .. 177* .. External Functions .. 178 LOGICAL LSAME 179 EXTERNAL LSAME 180* .. 181* .. External Subroutines .. 182 EXTERNAL XERBLA, ZDSCAL, ZHER, ZLACGV 183* .. 184* .. Intrinsic Functions .. 185 INTRINSIC DBLE, MAX, MIN, SQRT 186* .. 187* .. Executable Statements .. 188* 189* Test the input parameters. 190* 191 INFO = 0 192 UPPER = LSAME( UPLO, 'U' ) 193 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 194 INFO = -1 195 ELSE IF( N.LT.0 ) THEN 196 INFO = -2 197 ELSE IF( KD.LT.0 ) THEN 198 INFO = -3 199 ELSE IF( LDAB.LT.KD+1 ) THEN 200 INFO = -5 201 END IF 202 IF( INFO.NE.0 ) THEN 203 CALL XERBLA( 'ZPBSTF', -INFO ) 204 RETURN 205 END IF 206* 207* Quick return if possible 208* 209 IF( N.EQ.0 ) 210 $ RETURN 211* 212 KLD = MAX( 1, LDAB-1 ) 213* 214* Set the splitting point m. 215* 216 M = ( N+KD ) / 2 217* 218 IF( UPPER ) THEN 219* 220* Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m). 221* 222 DO 10 J = N, M + 1, -1 223* 224* Compute s(j,j) and test for non-positive-definiteness. 225* 226 AJJ = DBLE( AB( KD+1, J ) ) 227 IF( AJJ.LE.ZERO ) THEN 228 AB( KD+1, J ) = AJJ 229 GO TO 50 230 END IF 231 AJJ = SQRT( AJJ ) 232 AB( KD+1, J ) = AJJ 233 KM = MIN( J-1, KD ) 234* 235* Compute elements j-km:j-1 of the j-th column and update the 236* the leading submatrix within the band. 237* 238 CALL ZDSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 ) 239 CALL ZHER( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1, 240 $ AB( KD+1, J-KM ), KLD ) 241 10 CONTINUE 242* 243* Factorize the updated submatrix A(1:m,1:m) as U**H*U. 244* 245 DO 20 J = 1, M 246* 247* Compute s(j,j) and test for non-positive-definiteness. 248* 249 AJJ = DBLE( AB( KD+1, J ) ) 250 IF( AJJ.LE.ZERO ) THEN 251 AB( KD+1, J ) = AJJ 252 GO TO 50 253 END IF 254 AJJ = SQRT( AJJ ) 255 AB( KD+1, J ) = AJJ 256 KM = MIN( KD, M-J ) 257* 258* Compute elements j+1:j+km of the j-th row and update the 259* trailing submatrix within the band. 260* 261 IF( KM.GT.0 ) THEN 262 CALL ZDSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD ) 263 CALL ZLACGV( KM, AB( KD, J+1 ), KLD ) 264 CALL ZHER( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD, 265 $ AB( KD+1, J+1 ), KLD ) 266 CALL ZLACGV( KM, AB( KD, J+1 ), KLD ) 267 END IF 268 20 CONTINUE 269 ELSE 270* 271* Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m). 272* 273 DO 30 J = N, M + 1, -1 274* 275* Compute s(j,j) and test for non-positive-definiteness. 276* 277 AJJ = DBLE( AB( 1, J ) ) 278 IF( AJJ.LE.ZERO ) THEN 279 AB( 1, J ) = AJJ 280 GO TO 50 281 END IF 282 AJJ = SQRT( AJJ ) 283 AB( 1, J ) = AJJ 284 KM = MIN( J-1, KD ) 285* 286* Compute elements j-km:j-1 of the j-th row and update the 287* trailing submatrix within the band. 288* 289 CALL ZDSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD ) 290 CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD ) 291 CALL ZHER( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD, 292 $ AB( 1, J-KM ), KLD ) 293 CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD ) 294 30 CONTINUE 295* 296* Factorize the updated submatrix A(1:m,1:m) as U**H*U. 297* 298 DO 40 J = 1, M 299* 300* Compute s(j,j) and test for non-positive-definiteness. 301* 302 AJJ = DBLE( AB( 1, J ) ) 303 IF( AJJ.LE.ZERO ) THEN 304 AB( 1, J ) = AJJ 305 GO TO 50 306 END IF 307 AJJ = SQRT( AJJ ) 308 AB( 1, J ) = AJJ 309 KM = MIN( KD, M-J ) 310* 311* Compute elements j+1:j+km of the j-th column and update the 312* trailing submatrix within the band. 313* 314 IF( KM.GT.0 ) THEN 315 CALL ZDSCAL( KM, ONE / AJJ, AB( 2, J ), 1 ) 316 CALL ZHER( 'Lower', KM, -ONE, AB( 2, J ), 1, 317 $ AB( 1, J+1 ), KLD ) 318 END IF 319 40 CONTINUE 320 END IF 321 RETURN 322* 323 50 CONTINUE 324 INFO = J 325 RETURN 326* 327* End of ZPBSTF 328* 329 END 330